Question

Solve the following pair of equations by cross multiplication method.

4x − 5y + 7 = 0, 3x − 4y + 6 = 0

Answer 1

4x − 5y + 7 = 0     ...(1)

3x − 4y + 6 = 0     ...(2)

Let’s write equations in standard form:

a₁x + b₁y₁ + c₁ = 0

a₂x + b₂y + c₂ = 0

Here, they are in the form of,

a₁ = 4, b₁ = −5, c₁ = 7

a₂ = 3, b₂ = −4, c₂ = 6

Using the identity:

`x/(b_1c_2 - b_2c_1) = y/(c_1a_2 - c_2a_1) = 1/(a_1b_2 - a_2b_1)`

Now, substituting the values,

⇒ b₁c₂ − b₂c₁

= (−5)(6) − (−4)(7)

= −30 + 28

∴ b₁c₂ − b₂c₁ = −2

⇒ c₁a₂ − c₂a₁

= (7)(3) − (6)(4)

= 21 − 24

∴ c₁a₂ − c₂a₁ = −3

⇒ a₁b₂ − a₂b₁

= (4)(−4) − (3)(−5)

= −16 + 15

∴ a₁b₂ − a₂b₁ = −1

So, the value becomes,

`x/-2 = y/-3 = 1/-1`

Hence, finding x and y,

`x/-2 = 1/-1`

`x = (-2)(1/-1)`

`x = (-2)/-1`

∴ x = 2

`y/-3 = 1/-1`

`y = (-3)(1/-1)`

`y = (-3)/-1`

∴ y = 3

Thus, solving equations 4x − 5y + 7 = 0, 3x − 4y + 6 = 0 by cross multiplication method we get, x = 2 and y = 3.